Optimization Model and Solution Algorithm for the Design of Connected and Compact Nature Reserves

Ecosystem conservation is fundamental to guarantee the survival of endangered species and to preserve other ecological functions important for human systems (e.g., water). Planning land conservation increasingly requires a landscape approach to mitigate the negative impacts of spatial threats such as urbanization, agricultural development, and climate change. In this context, landscape connectivity and compactness are vital characteristics for the effective functionality of conservation areas. Connectivity allows species to travel across landscapes, facilitating the flow of genes across populations from different protected areas. Compactness measures the spatial dispersion of protected sites, which can be used to mitigate risk factors associated with species leaving and reentering the reserve. This research describes an optimization model for the design of conservation areas, while inducing connectivity and compactness. We use the Reocks index, a metric of compactness that maximizes the ratio of area of the selected patches to the area of their smallest circumscribing circle. Our model includes budget and minimum protected area constraints to reflect realistic financial and ecological requirements. The initial nonlinear model is reformulated into a mixed-integer linear program, which is solved using an adaptation of the Newtons method for problems with integer variables. We characterize an optimal solution and derive cuts to improve the model performance. We illustrate our results using real life landscapes with irregular patches.